Electron-ion thermal equilibration after spherical shock
collapse
The MIT Faculty has made this article openly available. Please share
how this access benefits you. Your story matters.
Citation
Rygg, J. R. et al. “Electron-ion thermal equilibration after
spherical shock collapse.” Physical Review E 80.2 (2009):
026403. (C) 2010 The American Physical Society.
As Published
http://dx.doi.org/10.1103/PhysRevE.80.026403
Publisher
American Physical Society
Version
Final published version
Accessed
Wed May 25 18:18:03 EDT 2016
Citable Link
http://hdl.handle.net/1721.1/51070
Terms of Use
Article is made available in accordance with the publisher's policy
and may be subject to US copyright law. Please refer to the
publisher's site for terms of use.
Detailed Terms
PHYSICAL REVIEW E 80, 026403 共2009兲
Electron-ion thermal equilibration after spherical shock collapse
J. R. Rygg,* J. A. Frenje, C. K. Li, F. H. Séguin, and R. D. Petrasso†
Plasma Science and Fusion Center, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
D. D. Meyerhofer‡ and C. Stoeckl
Laboratory for Laser Energetics, University of Rochester, Rochester, New York 14623, USA
共Received 3 March 2009; published 25 August 2009兲
A comprehensive set of dual nuclear product observations provides a snapshot of imploding inertial confinement fusion capsules at the time of shock collapse, shortly before the final stages of compression. The
collapse of strong convergent shocks at the center of spherical capsules filled with D2 and 3He gases induces
D-D and D- 3He nuclear production. Temporal and spectral diagnostics of products from both reactions are
used to measure shock timing, temperature, and capsule areal density. The density and temperature inferred
from these measurements are used to estimate the electron-ion thermal coupling and demonstrate a lower
electron-ion relaxation rate for capsules with lower initial gas density.
DOI: 10.1103/PhysRevE.80.026403
PACS number共s兲: 52.35.Tc, 52.25.Fi, 52.70.Nc, 52.57.⫺z
I. INTRODUCTION
Converging spherical shocks and electron-ion thermal
equilibration are basic physical processes 关1兴 of fundamental
importance for the design of high gain implosions in inertial
confinement fusion 共ICF兲 关2–4兴. Strong, spherically convergent shocks are formed by the rapid deposition of energy in
the form of lasers 共direct drive兲 or x rays 共indirect drive兲 on
the surface of a spherical capsule. Current “hot-spot” ICF
ignition designs include a sequence of up to four convergent
shocks that must be precisely timed to coalesce at the innershell surface so as to obtain maximal shell compression
关5,6兴, a necessity for high fusion gain. Other ignition designs
include the launching of a convergent shock into a compressed fuel assembly 关7兴. In both cases, ICF implosion design and performance is deeply affected by the speed and
heating of convergent shocks through ambient and compressed materials.
Shocks initially deposit thermal energy primarily in the
ions and the ensuing electron-ion thermal equilibration is one
of many related transport processes of concern for ICF modelers 关3,4兴. Recent theoretical 关8–10兴 and computational
关11–13兴 works have helped to clarify ambiguities in the
Landau-Spitzer energy equilibration rate 关14,15兴, which result from ad hoc cutoffs of logarithmic divergences in the
Coulomb collisional rates. Previous experimental and observational investigations of electron-ion thermal relaxation include the works of Celliers et al. 关16,17兴 and Laming et al.
关18兴 and new investigations are currently underway 关19,20兴.
This paper presents the first results of temporal and spectral measurements of products from two simultaneous
*Present address: Lawrence Livermore National Laboratory, 7000
East Avenue, Livermore, California 94550, USA.
†
Also at Laboratory for Laser Energetics, University of Rochester,
Rochester, New York 14623, USA.
‡
Also at Department of Mechanical Engineering, University of
Rochester, Rochester, New York 14623, USA; and Department of
Physics and Astronomy, University of Rochester, Rochester, New
York 14623, USA.
1539-3755/2009/80共2兲/026403共8兲
nuclear reaction types induced by the central collapse of convergent shocks. Nuclear measurements of some aspects of
shock collapse using a single nuclear product have been reported recently 关21–23兴. Observations of these products supply compelling information about the speed and heating of
the shocks and the state of the imploding capsule at the time
of shock collapse. In the experiments discussed here, this
occurs immediately before the onset of the deceleration
phase and the final stages of compression. The comprehensive picture of the central shocked gas provided by the dual
nuclear reaction measurements is used to evaluate electronion thermal equilibration in the plasma after shock collapse.
Section II describes the experimental setup and Sec. III
the experimental results. Various plasma parameters of the
central shocked gas are derived from the measurements in
Sec. IV. A brief review of electron-ion thermal equilibration
after shock heating is outlined in Sec. V and is applied to the
experimental observations in Sec. VI. Concluding remarks
are presented in Sec. VII.
II. EXPERIMENTAL SETUP
Direct-drive spherical capsule implosions were conducted
using the OMEGA laser system 关24兴, with 60 beams of ultraviolet 共351 nm兲 light in a 1 ns square pulse, a total energy
of 23 kJ, and full single-beam smoothing 关25兴. The resulting
1 ⫻ 1015 W / cm2 intensity was incident on capsules with diameters between 855 and 875 ␮m, plastic 共density
1.04 g / cm3兲 shell thicknesses 共⌬R兲 of 20, 24, or 27 ␮m,
and a flash coating of 0.1 ␮m of aluminum. The capsules
were filled with an equimolar 共by atom兲 mixture of D2 and
3
He gasses with a total fill pressure of 3.6 or 18 atm at 293
K, corresponding to initial gas mass densities 共␳0兲 of 0.5 and
2.5 mg/ cm3, respectively.
Three distinct primary nuclear reactions occur during capsule implosions with D2 and 3He fuel
026403-1
D + D → 3He + n,
共1兲
D + D → T + p,
共2兲
©2009 The American Physical Society
PHYSICAL REVIEW E 80, 026403 共2009兲
shock
1019
1017
1018
1
1.5
2
time (ns)
2.5
3
(b)
DD-p spectrum (108/MeV)
1.2
compression
D3He-p spectrum (108/MeV)
10
18
(a)
DD-n reaction rate (s-1)
D3He reaction rate (s-1)
RYGG et al.
compression
0.8
shock
0.4
0
birth
energy
9
12
15
E (MeV)
2
shock
1
0
18
(c)
birth
energy
0
1
E (MeV)
2
3
FIG. 1. Representative experimental observations of DD and D 3He nuclear products emitted at shock- and compression-bang times from
an implosion of a 24-␮m-thick CH capsule shell filled with 2.5 mg/ cm3 of D2- 3He gas 共OMEGA shot 38525兲. 共a兲 D 3He 共solid兲 and DD-n
共dotted兲 reaction-rate histories. 共b兲 D 3He proton spectrum. 共c兲 DD-proton spectrum.
D + 3He → 4He + p.
共3兲
The neutron 共1兲 and proton 共2兲 branches of the DD reaction
have nearly equal probabilities over temperatures of interest.
The D 3He reaction depends much more strongly on temperature due to the doubly-charged 3He reactant 关26兴. The
mean birth energies of D 3He and DD protons are 14.7 and
3.0 MeV, respectively.
Nuclear products were observed using the proton and neutron temporal diagnostics 共PTD and NTD兲 关22,27兴 to measure the D 3He and DD-n reaction histories, multiple wedgerange-filter proton spectrometers 关28兴 to measure the D 3He
proton yield and spectrum, and a magnet-based chargedparticle spectrometer 关28兴 to obtain the first measurements of
DD protons emitted at shock-bang time.
The D 3He reaction-rate history shows two distinct times
of nuclear production 关Fig. 1共a兲兴: “shock-burn” begins
shortly after shock collapse and ends near the beginning of
the deceleration of the shell and “compression-burn” lags
about 300 ps after shock burn, beginning near the onset of
shell deceleration and lasting approximately until the stagnation of the imploding shell 共Fig. 2兲. For ordinary D2- 3He
mixtures, the DD-n reaction rate during the shock burn is
below the diagnostic detection threshold.
The shock and compression components can often be distinguished in D 3He proton spectra 关Fig. 1共b兲兴 关21兴. The protons emitted during shock-burn experience relatively little
downshift 共⬃0.4 MeV兲 due to the low total capsule areal
density 共␳R兲 at that time. The shell continues to compress
after shock-burn ends and by the time of compression-burn,
the ␳R has increased enough to downshift D 3He protons by
several MeV.
The ␳R during shock-burn is low enough to allow nascent
3.0 MeV DD protons to escape the capsule 关Fig. 1共c兲兴, but
the DD protons are ranged out in the capsule during
compression-burn due to the higher capsule ␳R at that time.
Measurement of DD protons emitted during shock-burn provides a valuable and sole measurement of the DD shock
yield when the reaction rate is below the NTD threshold 共as
is often the case兲. Measurement of their downshift provides a
double check on the ␳R at shock-bang time inferred using the
D 3He proton spectra or the sole measurement in cases
where the shock component of the D 3He proton spectrum
cannot be separated from the compression component.
III. EXPERIMENTAL RESULTS
Measured shock-bang times and D 3He and DD-p shock
yields are shown in Fig. 3 as a function of ⌬R for implosions
of capsules with different ␳0 共see also Table I兲. The shockbang time 共ts兲 is the time of peak D 3He nuclear production
during the shock-burn phase, the shock-burn duration 共⌬ts兲 is
the full temporal width at half the maximum shock-burn production rate, the D 3He shock yield 共Y D 3He兲 includes only
the contribution from the higher-energy “shock” component
of the D 3He-proton spectrum, and the DD-p shock yield
共Y DD兲 includes only that part of the spectrum above the highenergy cutoff of protons accelerated from the shell 关29兴 关seen
at 0.8 MeV in Fig. 1共c兲兴. Figure 3 plots the mean and the
standard deviation of the mean for implosion ensembles of
each capsule configuration. Shot-by-shot tables of most of
the experimental results are available in Ref. 关30兴.
400
gas-shell
interface
300
nuclear
production
(a.u.)
shock front
radius
(P
Pm)
200
laser pulse
(a.u.)
100
0
0
0.5
1
1.5
2
2.5
time (ns)
FIG. 2. A representative 1D simulation shows the D 3He nuclear
reaction rate and trajectories of the gas-shell interface and the converging shock. Collapse of the converging shock induces nuclear
shock-burn about 300 ps before the compression-burn peak and
stagnation of the imploding shell. Reprinted with permission from
Ref. 关23兴. Copyright 2008, American Institute of Physics.
026403-2
ELECTRON-ION THERMAL EQUILIBRATION AFTER …
PHYSICAL REVIEW E 80, 026403 共2009兲
1.8
(a)
1.6
1.4
20
24
'R (Pm)
108
DD-p Shock Yield
(ns)
D3He Shock Yield
ts
107
27
2.5 mg/cm3
0.5 mg/cm3
(c)
(b)
20
24
'R (Pm)
108
107
27
20
24
'R (Pm)
27
FIG. 3. Experimental observations of 共a兲 shock-bang time, 共b兲 D 3He shock yield, and 共c兲 DD-p shock yield as a function of capsule shell
thickness for ensembles of capsules filled with 2.5 共triangles兲 or 0.5 mg/ cm3 共circles兲 of D 3He gas. In some cases, the error bars
共representing the standard error of the ensemble mean兲 are smaller than the marker size.
Experiments show that ts is linearly delayed with increasing ⌬R 关Fig. 3共a兲兴. No difference in ts was observed for
capsules with different ␳0. For capsules with the same ⌬R,
identical shocks should be generated in the shell with identical drive conditions 共as is approximately the case here兲 and
the shocks should break into the gas at the same time. Since
ts is independent of ␳0, we conclude that shocks of the same
speed are launched into the gas for implosions with the same
⌬R and drive.
Both D 3He and DD shock yields were observed to decrease for implosions of targets with thicker shells and lower
␳0. However, the expected yield reduction—due only to the
density dependence of the nuclear fusion rate—from high to
low ␳0 is 25, a much higher value than the observed reduction of between 3 and 5. This indicates that lower fill density
also results in reduced thermal coupling between ions and
electrons 共see Sec. V兲 so that the ion temperature, and consequently the nuclear fusion rate, remains high.
The average ion temperature 关31兴 at shock-bang time Tsi
can be inferred using the measured yields of the two different
nuclear reactions based on the ratio of their respective thermal reactivities 关26兴. This method has previously been used
to infer ion temperature during the compression burn by Li et
al. 关32兴 and Frenje et al. 关22兴. Figure 4 plots the Tsi inferred
by this method, showing higher Tsi for low ␳0 implosions.
The compression of the capsule at shock-bang time can be
quantified by the shock-burn-averaged 关31兴 areal density,
␳Rs. The areal density at shock time is of particular concern
in ICF because the value of ␳Rs sets the initial condition for
the final capsule compression to the stagnation ␳R, which in
turn is a fundamental metric of capsule assembly and is essential for ignition and efficient nuclear burn 关2–4兴. Experimentally, ␳Rs is inferred from the measured mean energy
downshift from the birth energy of DD protons 共␳Rs,DD兲 or
D 3He protons in the shock line 共␳Rs,D 3He兲, using a theoretical formalism to relate their energy loss to plasma parameters 关28,33兴. The inferred ␳Rs value is insensitive to the
exact parameter values assumed, particularly when using the
downshift of 14.7 MeV D 3He protons; a CH plasma density
of 3 g / cm3 and temperature of 0.3 keV were used to derive
the quoted ␳Rs values.
Substantial agreement is observed between ␳Rs inferred
from spectral results obtained using both DD and D 3He protons, as seen in Fig. 5 and Table I. Implosions with increasing ⌬R show an increase in ␳Rs due to the larger remaining
shell mass at shock time. On the basis of physical principles,
the contribution of the shell to the areal density ␳Rs,shell
should be only weakly dependent on the initial gas density ␳0
since the trajectory of the high-density shell will be almost
unaffected by the fill gas until the shell starts to decelerate
TABLE I. Mean and standard deviation of the mean of shock measurements with D 3He and DD protons for implosion ensembles of
different initial gas density 共␳0兲 and capsule shell thickness 共⌬R兲. Each ensemble consists of N 共NDD兲 implosions with D 3He 共DD-p兲
measurements. The D 3He ensemble includes shock-bang time 共ts兲, shock-burn duration 共⌬ts兲, D 3He shock yield 共Y D 3He兲, shock arealdensity 共␳Rs-d 3He兲, and inferred gas compression ratio 共␳s / ␳0关D 3He兴兲. The remaining quantities are from the DD ensemble, including the
DD-p shock yield 共Y DD兲, and the shock-burn-averaged ion temperature 共Tsi兲.
␳0
共mg/cc兲
⌬R
共␮m兲
Diam.
共␮m兲
N
共NDD兲
ts
共ps兲
⌬ts
共ps兲
Y D 3He
共⫻107兲
Y DD
共⫻107兲
Tsi
共keV兲
0.5
0.5
0.5
2.5
2.5
2.5
19.9
23.7
27.0
20.1
23.9
26.9
862
873
873
863
865
873
8共5兲
6共0兲
4共0兲
8共3兲
9共3兲
6共2兲
1470⫾ 16
1585⫾ 27
1731⫾ 39
1493⫾ 12
1591⫾ 12
1690⫾ 11
129⫾ 18
129⫾ 11
122⫾ 30
145⫾ 13
137⫾ 10
146⫾ 10
0.98⫾ 15%
0.48⫾ 9%
0.25⫾ 20%
3.09⫾ 7%
1.45⫾ 9%
1.44⫾ 18%
4.2⫾ 10%
7.7⫾ 0.7
14.1⫾ 13%
9.2⫾ 20%
026403-3
5.8⫾ 0.3
5.4⫾ 0.4
␳Rs-d 3He
共mg/ cm2兲
␳Rs-dd
共mg/ cm2兲
␳s / ␳0
关D 3He兴
8.3⫾ 0.7
9.8⫾ 0.4
12.0⫾ 0.9
8.2⫾ 1.0
9.1⫾ 0.7
9.4⫾ 1.2
9.3⫾ 0.6
10.0⫾ 0.7
11.1⫾ 1.0
␳s / ␳0
关DD兴
22⫾ 3
17⫾ 1
17⫾ 2
18⫾ 3
14⫾ 1
14⫾ 2
23⫾ 3
16⫾ 2
15⫾ 2
PHYSICAL REVIEW E 80, 026403 共2009兲
RYGG et al.
冉
␳s
␳Rs
=
␳0
␳R0,gas + f ␳R0,shell
10
0.5 mg/cc
8
Ts
6
(keV)
4
2
20
24
'R (Pm)
冉 冊
dm
I15
共g/cm2/s兲 = 2.6 ⫻ 105 4
dt
␭
27
FIG. 4. Shock-burn-averaged ion temperature vs ⌬R for two
different ␳0, calculated using the ratio of measured DD-p to D 3He
shock yields. Although shocked to the same initial ion temperature
at a given shell thickness, thermal coupling with electrons is weaker
in the low ␳0 implosions.
several hundred picoseconds after shock-bang time. As we
will see in Sec. IV, ␳Rs is dominated by the shell contribution
and should also be weakly dependent on ␳0. The data shown
in Fig. 5 and Table I are consistent with this viewpoint.
冉冊
␳Rs,gas = ␳R0,gas
Theoretical analysis suggests that converging shocks are
weakly unstable to initial asymmetries 关34兴; however, experiments have demonstrated that the nuclear observables are
highly robust to drive asymmetries 关23兴 and that the growth
of asymmetries due to hydrodynamic instabilities is insufficient to mix the shell with the fill gas at during the shockburn 关35兴. Thus, the behavior of the imploding capsule at the
time of shock-burn can be well described by a onedimensional 共1D兲, spherically symmetric model.
The shock-burn-averaged plasma density ␳s can be estimated from our measurements of the shock-burn-averaged
total areal density ␳Rs. Assuming thin shells and a spherically symmetric model of the implosion and invoking mass
conservation gives
U
UR (mg/cm2)
(a) D3He
(b) DD-p
10
5
0
20
24
'R (Pm)
27
20
24
'R (Pm)
27
FIG. 5. Shock-burn-averaged areal density ␳Rs vs ⌬R for D 3He
fills of 2.5 mg/ cm3 共triangles兲 and 0.5 mg/ cm3 共circles兲. ␳Rs is
inferred from the downshift of nascent 共a兲 14.7 MeV D 3He protons
and 共b兲 3 MeV DD protons from their birth energy. Markers show
mean and standard error.
1/3
,
共5兲
where I15 is the laser intensity in 1015 W / cm2 and ␭ is the
laser wavelength in microns. For these experiments, about
10 ␮m of the original shell is ablated during the laser pulse,
giving a volumetric compression ratio at shock time ␳s / ␳0 of
14–23 共see Table I兲. The inferred compression ratios are apparently equal for implosions with the same ⌬R but different
␳0, which is consistent with the expectation stated in the
previous section.
Using these values of the compression ratio, mass conservation can be used to estimate the areal density of the fuel at
shock time ␳Rs,gas,
IV. CHARACTERIZATION OF THE SHOCKED
GAS
15
共4兲
,
where ␳R0,gas and ␳R0,shell are the initial areal densities of the
gas and the shell before the implosion and f is the fraction of
the initial shell mass remaining after ablation of the outer
shell by the drive laser intensity. The mass ablation rate
dm / dt is 关3,4兴
2.5 mg/cc
0
冊
3/2
␳s
␳0
2/3
,
共6兲
which gives values of 0.15 and 0.6– 0.8 mg/ cm2, contributing 1%–2% and 6%–9% of the total ␳Rs for low and high ␳0,
respectively.
Simultaneous knowledge of the gas composition, density,
and temperature allows some basic plasma parameters to be
computed. For definiteness, the following discussion is restricted to the case of the ⌬R = 20 ␮m ensemble with high
共low兲 ␳0. The DD-inferred compression ratio, ⬃22, is the
same for all ␳0, but is slightly higher than the D 3He inferred
compression ratio, 18, for high ␳0. The average of these
methods gives a compression ratio ␳s / ␳0 = 20, which will be
used for both ensembles. In this case, at shock-bang time, the
mass density ␳s = 50共10兲 mg/ cm3, the electron density ne
= 18共3.6兲 ⫻ 1021 cm−3, and the Fermi energy E f
= ប2共3␲2ne兲2/3 / 2me = 2.5共0.86兲 eV, where ប is the reduced
Planck constant and me is the electron mass.
As will be shown in Sec. V, the electron temperature Te
averaged over shock-burn is 2.0 共0.73兲 keV, which establishes that the electrons can be treated as nondegenerate: the
electron degeneracy parameter ⌰ = Te / E f = 800共850兲 Ⰷ 1.
Both the electron and ion temperatures are much higher than
the final ionization energies of atomic D and 3He 共D: 13.6
eV, 3He : 54.4 eV兲, so the gas is a fully ionized plasma.
The pressure in a nondegenerate fully ionized plasma is
given by the ideal kinetic gas pressure, P = 共neTe + niTi兲
= 17共3.4兲 TPa. As temperatures in this paper are expressed
in energy units, Boltzmann’s constant kB has been suppressed. The plasma parameter, related to the number of par3/2
ticles in a Debye sphere, is 共␧0Te / e2n1/3
e 兲 = 1900共950兲 Ⰷ 1.
␧0 is the permittivity of free space and e is the fundamental
charge.
026403-4
ELECTRON-ION THERMAL EQUILIBRATION AFTER …
PHYSICAL REVIEW E 80, 026403 共2009兲
2.5
The Coulomb logarithm, ln ⌳ = ln共bmax / bmin兲, is important
for many plasma transport properties, including thermal
equilibration, but there is some variation in the precise
impact-parameter cutoffs bmax and bmin 关13,15兴. Here, we use
the value of ln ⌳ given by Ref. 关10兴 in the nondegenerate
limit
冉 冊
ln ⌳ = ln
Te
− 1.8283,
ប␻ pe
If a strong, nonradiating shock propagating at speed us
through a uniform ideal gas is sufficiently strong to fully
ionize the gas 共as is the case here兲, it will distribute thermal
energy among the electron and ion species according to their
masses m j, such that the immediate post-shock temperatures
T0j are 共e.g., see Ref. 关36兴兲
3
m jus2 ,
16
共8兲
where j = e , i for electrons and ions.
The large mass difference between the ions and electrons
共⬃4600 for the equimolar D-3He mixture considered here兲
endows each species with widely different initial temperatures, but otherwise depends only on the shock speed. The
electron and ion temperatures 共Te and Ti兲 relax over time to a
final equilibrium temperature T f as energy is exchanged
through Coulomb collisions. In the absence of thermal conduction, the sum of Te and Ti is constrained by energy conservation according to their relative heat capacities
Ti + ZTe = T0i + ZT0e = 共1 + Z兲T f ,
U
1
0.5
U/5
Te
0
0.001
0.01
0.1
共10兲
where ␶ei is the electron-ion thermal equilibration time constant 关37,38兴 and is temperature dependent
1
t / Wf
10
FIG. 6. Electron-ion thermal equilibration for Z = 1.5. Ion 共bold
lines兲 and electron 共thin lines兲 temperatures approach to within a
few percentage of their equilibrium value by time ␶ f . Thermal relaxation for plasmas with 1/5 of the reference mass density takes
approximately 5 times as long 共dotted lines兲.
冉冊
3/2
,
共11兲
where ␶ f is the density-dependent coupling time constant at
the equilibrium temperature 关4,13,15兴,
␶f =
冉 冊
4␲␧0
2
8冑2␲ e
3
2
m2i
T3/2
f
.
␳
ln
⌳f
Z2m1/2
e
共12兲
Here, ␳ is the mass density and ln ⌳ f is the Coulomb logarithm given by Eq. 共7兲 with Te → T f . The small logarithmic
dependence of ln ⌳ f on temperature has been neglected in
Eq. 共11兲.
Using Eqs. 共9兲 and 共11兲, Eq. 共10兲 becomes
dTe 共1 + Z兲T f 1 − Te/T f
.
=
共Te/T f 兲3/2
dt
␶f
共13兲
Replacing Te / T f → T and t / ␶ f → t, the integral representation
is
共1 + Z兲
冕 冕
dt =
T3/2dT
,
1−T
共14兲
which is analytically integrable
共9兲
where Z = 1.5 is the average ion atomic number. Note that
ZT0e Ⰶ T0i.
The rate of temperature equilibration is usually expressed
as the ratio of the temperature difference over a characteristic
time 关1,15兴
dTe Ti − Te
=
,
dt
␶ei
T / Tf
␶ei
Te
=
␶f
Tf
V. THERMAL EQUILIBRATION
T0j =
1.5
共7兲
where ␻ pe = 共e2ne / ␧0me兲1/2 is the electron plasma frequency.
For the gas at shock time, Eq. 共7兲 gives ln ⌳ = 6.2共6.0兲.
It should be emphasized that this characterization of the
shocked gas completely ignores many attributes of this
highly dynamic and nonuniform system, including steep
temperature and density gradients, nonthermal velocity components, and rapid temporal evolution. However, describing
the plasma in this “shock-averaged” manner 关31兴 offers valuable information about the state of the imploding capsule
immediately before the onset of deceleration phase, both as
an initial condition of and in contrast to the compression
burn. In addition, comparison of the shock states with different ␳0 allows the value of the electron-ion thermal equilibration rate to be inferred experimentally.
Ti
2
共1 + Z兲t = 2 tanh−1关冑T兴 −
2
冑T共T + 3兲.
3
共15兲
Figure 6 is a plot of this relation for Z = 1.5.
VI. MEASURING THERMAL EQUILIBRATION
The initial ion temperature T0i imparted by the shock in
Eq. 共8兲 depends only on mi and us. The experimental results
reported above are consistent with the independence of us on
the initial gas density ␳0. Since the same gas composition
was used for all experiments, this implies that the converging
026403-5
PHYSICAL REVIEW E 80, 026403 共2009兲
RYGG et al.
shocks launched into capsules with different ␳0 nonetheless
are heated to the same T0i. These situations have coupling
rates different by a known factor, since the equilibrium time
constant ␶ depends on ␳s.
T0i can be estimated using the finite difference form of
Eq. 共10兲,
⌬Te Ti − Te
.
=
⌬t
␶ei
10
8
Ti
共16兲
(keV)
6
50 mg/cc
4
experimental duration and Ti uncertainty
Using Eq. 共9兲 and assuming T0e is negligible, ⌬Te = Te = 共T0i
− Ti兲 / Z and
2
T0i − Ti 共1 + Z兲Ti − T0i
=
.
⌬t
␶ei
0
共17兲
If Ti reaches the measured shock-burn-averaged ion temperature Tsi after ⌬t equal to half the burn duration ⌬ts, then all
quantities are known except for Ti0 and ␶ei. These values
have a known relationship for high and low ␳0, so the two
sets of measurements are combined to solve for T0i. Using
indices 1 and 2 for high and low ␳0, we obtain
⌬ts2 T0i − Ts1 ␶ f2 共1 + Z兲Ts1 − T0i
=
.
⌬ts1 T0i − Ts2 ␶ f1 共1 + Z兲Ts2 − T0i
共18兲
From Table I, ⌬ts2 / ⌬ts1 = 0.89 and from Eq. 共12兲, ␶ f2 / ␶ f1
= 4.5. Defining k = 共⌬ts2 / ⌬ts1兲共␶ f1 / ␶ f2兲 and expanding gives a
quadratic equation for T0i,
k共T0i − T1兲关共1 + Z兲T2 − T0i兴 = 共T0i − T2兲关共1 + Z兲T1 − T0i兴,
共19兲
with coefficients
a = 共1 − k兲,
b = k关共1 + Z兲T2 + T1兴 − 共1 + Z兲T1 − T2 ,
c = 共1 − k兲共1 + Z兲T1T2 .
共20兲
Using the values from Table I, solutions for T0i at 12.7
and 8.8 keV are obtained. The 12.7 keV solution is rejected
as too high compared to observations of Tsi 关39兴. The 8.8
keV solution corresponds to an equilibrium temperature T f
= 3.5 keV. This is substantially lower than our measured Tsi
of 5.8 共7.7兲 keV for high 共low兲 ␳0, indicating that both implosion types are far from thermal equilibrium during the
shock burn.
With this shock-burn-averaged estimate of T0i, Eq. 共9兲 and
the measurements of Tsi are used to estimate the shock-burnaveraged electron temperature, giving Tse = 2.0共0.73兲 keV, as
stated in Sec. IV. In that section, we also estimated the
plasma density ␳s, which with Tse can be used to calculate
the shock-burn-averaged ␶ei by Eqs. 共11兲 and 共12兲, giving
characteristic times of 410 共470兲 ps 关40兴. These coupling
times are longer than the shock-burn duration, indicating that
both implosion types have a large temperature difference at
the end of the shock burn.
The initially surprising similarity of the characteristic
time constants for high and low ␳s can be explained by considering that the electrons in the high ␳s implosion have al-
10 mg/cc
equilibrium temperature
0
50
100
t (ps)
150
200
FIG. 7. Ion temperature relaxation for D 3He plasmas of density
␳s = 50 共solid兲 and 10 共dotted兲 mg/ cm3. The curves represent the
temperature equilibration starting at an initial ion temperature T0i
= 8.8 keV 共corresponding to T f = 3.5 keV兲. The width of the gray
boxes represents the average measured shock-burn duration and the
height represents the 1-sigma confidence interval of the experimental shock-burn-averaged ion temperature, Tsi. Compression-burn
overwhelms the shock-burn dynamics starting ⬃200 ps after shock
collapse.
ready absorbed much more thermal energy, thereby increasing the time constant as it takes more collisions to heat them
further. More illustrative of the difference in the equilibration
rates are the ion temperature relaxation curves according to
Eq. 共15兲, plotted in Fig. 7 for high and low ␳s from an initial
temperature T0i = 8.8 keV. From the figure, it is evident that
the slopes of the two relaxation curves are similar except for
very near t = 0 when the high ␳s plasma undergoes rapid
equilibration.
Also shown in Fig. 7 are the measured burn duration and
burn-averaged ion temperature Tsi for implosions with high
and low ␳0. The temperature relaxation curves calculated in
the simple model are consistent with the average ion temperature inferred from nuclear yield measurements. However, it should again be noted that the central gas during the
shock burn is far from the uniform plasma assumed here, as
the shock reflected after collapse will heat the fuel to different initial temperatures at different times as it propagates
outwards toward the incoming shell.
VII. CONCLUSIONS
In summary, nuclear production induced by the collapse
of strong, spherically convergent shocks was observed using
temporal and spectral measurements of products from two
distinct, simultaneous nuclear reaction processes. These dual
nuclear shock-burn measurements, hitherto unavailable, create a comprehensive description of the state of the implosion
immediately after shock collapse time—with gas ion temperatures, gas electron densities, and total areal densities at
shock-bang time near 6 keV, 1022 e− / cm3, and 10 mg/ cm2,
respectively.
026403-6
ELECTRON-ION THERMAL EQUILIBRATION AFTER …
PHYSICAL REVIEW E 80, 026403 共2009兲
The extensive information provided by these shock-burn
measurements demonstrate that the ions and electrons are far
from thermal equilibrium at the end of the shock burn—
particularly so for plasmas of lower density. Ion temperature
relaxation curves are calculated with a theoretical thermal
equilibration model 关10兴 using plasma parameters inferred
from shock-yield-averaged measurements. These calculated
ion temperature curves—which assume the plasma to be otherwise static and uniform—are consistent with the observed
temperatures, despite the dynamic and highly nonuniform
plasma state.
Future experiments could explore thermal equilibration in
denser plasmas using simple modifications of the methods
described herein. For example, the shell could be filled to
larger initial density, either with cryogenically cooled gas or
alternatively with 3He-wetted, deuterated-plastic foam. Plasmas at much higher areal densities can be investigated with
this technique using D 3He protons and DD neutrons if the
compression component can be suppressed or significantly
关1兴 Ya. B. Zel’dovich and Yu. P. Raizer, Physics of Shock Waves
and High-Temperature Hydrodynamic Phenomena 共Dover
Publications, New York, 2002兲.
关2兴 J. Nuckolls et al., Nature 共London兲 239, 139 共1972兲.
关3兴 J. D. Lindl, Inertial Confinement Fusion 共Springer-Verlag,
New York, 1999兲.
关4兴 S. Atzeni and J. Meyer-Ter-Vehn, The Physics of Inertial Fusion 共Oxford University, Oxford, 2004兲.
关5兴 D. H. Munro et al., Phys. Plasmas 8, 2245 共2001兲.
关6兴 R. L. McCrory et al., Nucl. Fusion 41, 1413 共2001兲.
关7兴 R. Betti, C. D. Zhou, K. S. Anderson, J. L. Perkins, W.
Theobald, and A. A. Solodov, Phys. Rev. Lett. 98, 155001
共2007兲.
关8兴 D. O. Gericke, M. S. Murillo, and M. Schlanges, Phys. Rev. E
65, 036418 共2002兲.
关9兴 L. S. Brown, D. L. Preston, and R. L. Singleton, Jr., Phys. Rep.
410, 237 共2005兲.
关10兴 L. S. Brown and R. L. Singleton, Jr., Phys. Rev. E 76, 066404
共2007兲.
关11兴 M. S. Murillo and M. W. C. Dharma-wardana, Phys. Rev. Lett.
100, 205005 共2008兲.
关12兴 B. Jeon, M. Foster, J. Colgan, G. Csanak, J. D. Kress, L. A.
Collins, and N. Gronbech-Jensen, Phys. Rev. E 78, 036403
共2008兲.
关13兴 G. Dimonte and J. Daligault, Phys. Rev. Lett. 101, 135001
共2008兲.
关14兴 L. D. Landau, Phys. Z. Sowjetunion 10, 154 共1936兲.
关15兴 L. Spitzer, Physics of Fully Ionized Gases 共Wiley, New York,
1962兲.
关16兴 P. Celliers, A. Ng, G. Xu, and A. Forsman, Phys. Rev. Lett.
68, 2305 共1992兲.
关17兴 A. Ng, P. Celliers, G. Xu, and A. Forsman, Phys. Rev. E 52,
4299 共1995兲.
关18兴 J. M. Laming et al., Astrophys. J. 472, 267 共1996兲.
关19兴 J. M. Taccetti et al., J. Phys. A 39, 4347 共2006兲.
关20兴 J. J. Angulo Gareta and D. Riley, High Energy Density Phys.
delayed, perhaps by using thicker shells, greater energy, or
shaped laser pulses. The application of one or more of these
modifications would further enhance the e-i thermal coupling
and push the investigation of temperature equilibration toward the challenging strongly coupled plasma regime.
ACKNOWLEDGMENTS
The authors express their gratitude to the OMEGA engineers and operations crew who supported these experiments.
This work was supported in part by the U.S. Department of
Energy Office of Inertial Confinement Fusion 共Grant No.
DE-FG03-03NA00058兲; the Laboratory for Laser Energetics
共Subcontract No. 412160-001G兲 under Cooperative Agreement No. DE-FC52-92SF19460, University of Rochester;
New York State Energy Research and Development Authority; and performed in part under the auspices of the U.S.
Department of Energy by Lawrence Livermore National
Laboratory under Contract No. DE-AC52-07NA27344.
关21兴
关22兴
关23兴
关24兴
关25兴
关26兴
关27兴
关28兴
关29兴
关30兴
关31兴
关32兴
关33兴
关34兴
关35兴
关36兴
关37兴
关38兴
026403-7
2, 83 共2006兲.
R. D. Petrasso et al., Phys. Rev. Lett. 90, 095002 共2003兲.
J. A. Frenje et al., Phys. Plasmas 11, 2798 共2004兲.
J. R. Rygg et al., Phys. Plasmas 15, 034505 共2008兲.
T. R. Boehly et al., Opt. Commun. 133, 495 共1997兲.
S. Skupsky et al., Phys. Plasmas 6, 2157 共1999兲.
H.-S. Bosch and G. M. Hale, Nucl. Fusion 32, 611 共1992兲.
R. A. Lerche et al., Rev. Sci. Instrum. 66, 933 共1995兲.
F. H. Séguin et al., Rev. Sci. Instrum. 74, 975 共2003兲.
Protons from the shell material are accelerated by electrostatic
fields while the laser pulse illuminates the capsule. These fields
have decayed well before the time of nuclear production, several 100 ps after the end of the pulse, so do not affect nuclear
product spectra. See also D. G. Hicks et al., Phys. Plasmas 8,
606 共2001兲.
J. R. Rygg, Ph.D. thesis, Massachusetts Institute of Technology, 2006.
Note that the shock-burn averages discussed in this paper are
not direct spatiotemporal averages, but are weighted by the
DD-p and D 3He nuclear production rates, which have strong
temperature and density dependence.
C. K. Li et al., Phys. Plasmas 7, 2578 共2000兲.
C. K. Li and R. D. Petrasso, Phys. Rev. Lett. 70, 3059 共1993兲.
J. H. Gardner et al., J. Fluid Mech. 114, 41 共1982兲.
J. R. Rygg, J. A. Frenje, C. K. Li, F. H. Seguin, R. D. Petrasso,
V. Y. Glebov, D. D. Meyerhofer, T. C. Sangster, and C. Stoeckl, Phys. Rev. Lett. 98, 215002 共2007兲.
R. P. Drake, High Energy Density Physics 共Springer, New
York, 2006兲.
Note that the parameter ␶ei—commonly called the electron-ion
equilibration time constant—is not actually constant in time
for the large temperature differences considered here.
The large ion-electron mass ratio makes collisions inefficient
for exchanging energy between the two species, so generally
the individual species will equilibrate on a much faster time
scale than the relaxation between the species.
PHYSICAL REVIEW E 80, 026403 共2009兲
RYGG et al.
关39兴 The 12.7 keV root gives T f = 5.1 keV and a characteristic time
of 1270 共3660兲 ps. At peak shock-burn, the corresponding ion
temperature would be 8.8 共10.4兲 keV.
关40兴 More appropriate to describe the overall shape of the relaxation curve is ␶ f , the equilibration time constant at the equilib-
026403-8
rium temperature described in Eq. 共12兲: ␶ f = 880共3900兲 ps for
high 共low兲 ␳0. However, since ⌬ts Ⰶ ␶ f , ␶ f is not suitable for
describing the coupling time characteristic of the plasma during shock burn.